Gauge-independent $\overline{MS}$ renormalization in the 2HDM

We present a consistent renormalization scheme for the CP-conserving Two-Higgs-Doublet Model based on $\overline{MS}$ renormalization of the mixing angles and the soft-$Z_2$-symmetry-breaking scale $M_{sb}$ in the Higgs sector. This scheme requires to treat tadpoles fully consistently in all steps of the calculation in order to provide gauge-independent $S$-matrix elements. We show how bare physical parameters have to be defined and verify the gauge independence of physical quantities by explicit calculations in a general $R_{\xi}$-gauge. The procedure is straightforward and applicable to other models with extended Higgs sectors. In contrast to the proposed scheme, the $\overline{MS}$ renormalization of the mixing angles combined with popular on-shell renormalization schemes gives rise to gauge-dependent results already at the one-loop level. We present explicit results for electroweak NLO corrections to selected processes in the appropriately renormalized Two-Higgs-Doublet Model and in particular discuss their scale dependence.


Introduction
The discovery of a Higgs boson at the Large Hadron Collider (LHC) [1,2] was a tremendous success for elementary particle physics. The Higgs boson is now a central object of intense research in both experiment and theory in order to determine its properties precisely. In particular, it is interesting to investigate whether the Higgs boson belongs to the Standard Model (SM) or whether it is part of a more general theory. In this context, models with additional Higgs bosons are of special interest. An extended Higgs sector can contribute to solve open problems in particle physics, like for example the question of the origin of the matter-antimatter asymmetry in the universe or the nature of dark matter.
At the LHC, detectable differences between the SM and a theory with an extended Higgs sector can be small. Therefore, accurate theory predictions are strongly desirable and in turn require the knowledge of higher-order corrections. QCD corrections essentially dress the basic electroweak (EW) interactions of the Higgs boson and do not fundamentally change by adding additional Higgs bosons to the theory. Electroweak corrections, on the other hand, can significantly modify the predictions for physical observables, like cross sections and partial decay widths, through an extended Higgs sector. For this reason, we dedicate special attention to the calculation of next-to-leading order (NLO) EW corrections to Higgs production. This requires a renormalization of the new physical parameters and fields of the extended Higgs sector.
Within this work, we consider in particular the CP-conserving 2HDM of type II with a softly broken Z 2 symmetry [3,4]. The Higgs sector of this model depends on four physical mass parameters, the masses M H l and M H h of the light and heavy, neutral, scalar Higgs bosons, the mass M Ha of the pseudo-scalar Higgs boson, and the mass M H ± of the charged Higgs boson. In addition, there are two mixing angles, α and β, as well as the soft-Z 2 -breaking scale M sb . The renormalization of the 2HDM of type II has been discussed in the context of supersymmetry (see e.g. Refs. [5][6][7]) and also in the general case [8][9][10][11]. A reasonable renormalization scheme should fulfil the following three conditions [6]: it should lead to gauge-independent physical counterterms, it should be numerically stable, i.e. the size of the higher-order corrections should be moderate, and it should preferably be defined in a process-independent way.
The masses of the Higgs bosons, vector bosons and fermions are naturally renormalized in the classical on-shell scheme, which is straightforward to apply. In contrast, for the parameters α, β, and M sb there is no natural renormalization scheme. As long as these parameters are unknown, it is difficult to identify processes that can be measured precisely and that would allow to extract the values of these parameters accurately and in a stable way. Moreover, processdependent renormalization can lead to unnaturally large corrections for the predictions of other processes (see e.g. Ref. [11]) or require to artificially split off IR singularities [6]. Motivated by studies on the renormalization of the quark-mixing matrix in the SM [12], process-independent renormalization of the mixing angles α and β based on the field renormalization constants of the Higgs fields have been proposed [9,10,13]. Taking these recipes naively leads, however, to gaugedependent renormalization of the mixing angles [14]. Therefore, it has been suggested [11,15] to render these methods gauge independent by applying the pinch technique [16,17]. This, however, merely trades the gauge dependence for a dependence on the prescription intrinsic in the pinch technique.
We consider it favourable to stick to a renormalization scheme that does not depend on particular conventions and propose to use the MS renormalization scheme which is by default used in perturbative QCD. MS renormalization is simple to apply and to implement. In addition, it leads to a residual scale dependence that helps to estimate the uncertainty caused by unknown higher-order corrections via a suitable scale variation. If the perturbative corrections become 1 large along with the scale dependence, this signals the onset of the non-perturbative regime where the Higgs sector becomes strongly coupled. This will be illustrated in this work.
The use of an MS renormalization scheme, however, requires particular attention in the proper treatment of the Higgs tadpoles. Tadpoles have no physical meaning and drop out in properly calculated S-matrix elements. If all parameters of the theory can be renormalized onshell, as in the SM, (some) tadpoles can be consistently omitted to simplify practical calculations. However, if parameters are renormalized in the MS scheme, it becomes crucial to define all bare physical parameters and counterterms, like for instance the mass counterterms of the EW gauge bosons or Higgs bosons, in a gauge-independent way. This in turn requires a consistent treatment of tadpoles. Within the 2HDM, we show that a careless treatment of tadpoles, as often used in the literature, in combination with MS renormalization of the mixing angles α and β yields gauge-dependent results already at the one-loop level.
A consistent treatment of tadpoles has been formulated for the SM by Fleischer and Jegerlehner in Ref. [18]. In this paper, we generalize this scheme, which we dub FJ Tadpole Scheme, and apply it to the 2HDM in order to allow for a consistent gauge-independent MS renormalization of the mixing angles α and β as well as the soft-breaking scale M sb . We stress that the FJ Tadpole Scheme as introduced here provides a consistent universal description of tadpoles in arbitrary theories with spontaneous symmetry breaking (SSB), such as general multi-Higgs models.
The outline of this paper is as follows. In Section 2 we discuss the role of the Higgs tadpoles for the proper definition of gauge-independent, physical parameters and counterterms for a general Higgs sector and recapitulate the use of the FJ Tadpole Scheme within the SM. Our notation and conventions for the 2HDM are introduced in Section 3. Section 4 is devoted to the extension of the FJ Tadpole Scheme to the 2HDM and the formulation of the renormalization conditions. In Section 5 we discuss the gauge dependence of popular schemes and relate these schemes to the FJ Tadpole Scheme. In Section 6 we demonstrate the applicability of the FJ Tadpole Scheme by employing it in NLO calculations of Higgs-boson production processes in the 2HDM. In particular, we discuss Higgs-boson production in gluon fusion as well as Higgsboson production through Higgs strahlung. Finally, in Section 7 we close with our conclusions. In the Appendices we present results for the Higgs-tadpole counterterms in the 2HDM, proof the gauge dependence of the MS renormalization of β in popular tadpole schemes, and illustrate the tadpole contributions to the two-loop Higgs-boson self-energy.

The role of tadpoles and gauge dependence
Before discussing the renormalization of the 2HDM, we examine the tadpole renormalization in theories with an extended Higgs sector in general, and we revisit the treatment of tadpoles in the SM. As tadpole contributions are gauge dependent (see e.g. Refs. [19,20] or App. A), their gauge-dependent parts have to cancel in any physical quantity. This cancellation is always given when a physical renormalization scheme such as the on-shell scheme is applied, which is usually the case in the EW SM. For Beyond-Standard-Model (BSM) theories, on the other hand, on-shell renormalization of parameters may introduce a process dependence and/or lead to unnaturally large NLO corrections. Hence, it is preferable to renormalize new parameters in the MS scheme, at least until first evidence of the BSM theory allows for a meaningful definition of a processdependent renormalization scheme. In this case, the treatment of tadpole counterterms requires some care to warrant the gauge independence of S-matrix elements: if MS counterterms are used, it is essential to ensure that all counterterms of physical parameters are gauge independent. This requires a gauge-independent definition of bare physical parameters which, in particular, must not employ shifted vacuum expectation values (vevs) in the presence of SSB. We show how this can be achieved in general and in a systematic way. In Section 2.1, we discuss a proper definition of the vevs for a general Higgs sector. The treatment of tadpoles closely follows the arguments which have been presented by Fleischer and Jegerlehner in App. A of Ref. [18] for the SM. We refer to this scheme as the FJ Tadpole Scheme in the following.
The tadpole counterterms are absent at tree level. At higher orders in perturbation theory, they can be used to remove explicit tadpole contributions, which simplifies practical calculations. We show that the FJ Tadpole Scheme is gauge independent regardless of whether tadpoles are removed by a consistent renormalization or taken into account explicitly. This implies that any physical quantity is independent of the (consistent) renormalization of tadpoles.

The FJ Tadpole Scheme for a general Higgs sector
The bare Higgs Lagrangian is defined in terms of bare fields Φ i,B and bare parameters c j,B , with i = 1, . . . , l and j = 1, . . . , k. We stress that c j,B are the theory-defining parameters, i.e. the bare parameters of the original Lagrangian with unbroken gauge symmetry. After spontaneous symmetry breaking, the neutral scalar components ϕ i,B of the Higgs multiplets obtain vevs v i,B + ∆v i , and the Lagrangian can be written as where the shifts of the vevs ∆v i are introduced for later convenience. The v i,B are chosen in such a way that the vevs of the shifted fields ϕ i,B ϕ i,B = 0 + t i (∆v 1 , . . . , ∆v l ) + higher-order corrections, (2.3) vanish at tree level, where ∆v = 0. Here ∆v = 0 is short for ∆v 1 = 0, . . . , ∆v l = 0. Since the v i,B are thus defined to minimize the bare scalar potential, they are directly given in terms of the bare (theory-defining) parameters of the Lagrangian [see (2.32) for the SM]. The tadpole counterterm t i is defined by the expression obtained by taking the derivative of the Lagrangian with respect to the field ϕ i , setting all fields to zero and keeping only the linear and higher-order terms in ∆v i . This can be generalized to tadpole counterterms of n-point functions. To this end, we define the tadpole Lagrangian ∆L in the FJ Tadpole Scheme, which gives rise to all the tadpoles in the theory, as Then, the tadpole counterterm is defined by the expression where the field ϕ i can be any scalar in the theory. The tadpole counterterms to two-point functions are given by derivatives of the Lagrangian with respect to ϕ i and ϕ j , i.e.
t ij (∆v 1 , . . . , ∆v l ) ≡ ∆L ij (∆v 1 , . . . , ∆v l ; . . .) with ∆L ij := where the fields ϕ i and ϕ j can be scalars, vector bosons or fermions. Analogously, we obtain the tadpole counterterm to the interaction of three fields where the fields ϕ i , ϕ j and ϕ k can only be scalars or vector bosons in renormalizable quantum field theories. Tadpole counterterms to scalars arise from the Higgs potential, while the tadpole counterterms involving vector bosons and fermions originate from the kinetic terms and the Yukawa terms, respectively, of the Lagrangian of the theory. The one-particle irreducible (1PI) tadpole loop corrections are given by with the contributions 1 at tree level (0), one loop (T (1) i ), two loops (T (2) i ), and the hatched graph denoting the sum of all 1PI tadpole graphs. Note that T Choosing the shifts of the vevs as ∆v = 0 implies that all tadpole counterterms t i vanish, and ϕ i,B = 0 holds at tree level. Already at one-loop order, the bare fields receive a nonvanishing vev due to the tadpole loop corrections in Eq. (2.8). In the following, we illustrate the tadpole renormalization for general two-point functions. We define the self-energy Σ ii (q 2 ) of a field i as the higher-order (beyond tree-level) contributions to the inverse connected 2-point function and denote the corresponding 1PI contributions by Σ 1PI ii . We indicate renormalized functions by a hat. The renormalized self-energy at one-loop order can be written in terms of 1PI graphs as followŝ where the subscript R indicates renormalized 1PI graphs. For the two-loop contributions we obtainΣ

4
where we suppressed the summation over tadpoles of different fields for simplicity. In the FJ Tadpole Scheme the counterterm corresponding to the two-point irreducible part reads where the tadpole counterterm t ii is defined in Eq. (2.6), δm 2 i is the mass counterterm, and δZ i is the field-renormalization counterterm. In addition, the renormalized tadpole contributionT ii with the sum over all scalar fields has to be taken into account. The physical mass is defined as the zero of the full 2-point function. 2 Accordingly in the on-shell scheme the mass counterterm δm 2 i is obtained by requiring the renormalized on-shell self-energyΣ ii (m 2 i ), as defined in Eq. (2.9), to vanish. One can show that the resulting counterterm is gauge independent, for example, by means of extended BRST symmetry (following the proof for W bosons in Ref. [19] in the R ξgauge). We have verified that the gauge parameters ξ Z and ξ W [see Eq. (5.2)] cancel for mass counterterms in the SM and the 2HDM by explicit calculations in the R ξ gauge.
As stated above, both t ii and t i vanish for ∆v = 0. Therefore, the non-renormalized sum of the tadpole diagrams T ii , i.e. the third contribution in Eq. (2.9), has to be taken into account. To avoid calculating these contributions, we can allow ∆v = 0 in Eq. (2.2) and relate ∆v to the loop tadpole corrections. More precisely, we choose the shift in the vev ∆v such that the fields ϕ do not develop a vev to any order in perturbation theory, which is equivalent to setting all renormalized tadpole contributionsT i of the theory to zero. The shifts ∆v are determined by solving the non-linear equations with t i defined in Eq. (2.5). This equation is solved order by order in perturbation theory upon using the perturbative expansion of ∆v, ∆v i = ∆v (1) i + ∆v (2) i + . . . , (2.14) and the perturbative expansion of the tadpole contributions (2.8).
As an important consequence of the consistent inclusion of tadpole contributions, as in Eqs. (2.10) and (2.11), connected Green's functions do not depend on a particular choice of ∆v. To proof this, we consider the generating functional of Green's functions Z[j] defined using the Lagrangian (2.2) with ∆v = 0 and the generating functional Z [j] based on Eq. (2.2) with an arbitrary ∆v = 0. Restricting ourselves for simplicity to the case with only one Higgs field ϕ, the two functionals can be related by a field redefinition ϕ → ϕ − ∆v in Z [j] using the invariance of the path integral measure under a constant shift.
Consequently, it follows for the connected Green's functions In the presence of mixing this requires an appropriate renormalization of the mixing energies Σij, i = j.

5
Note that this does not imply that the vertex functions are the same. The connection between the vertex functions in both formulations is given by andφ =φ − ∆v. (2.18) Treating ∆v perturbatively, the n-point vertex functions are related by However, Eq. (2.16) states that the tadpole counterterm dependence originating from Γ ∆ cancels in connected Green's functions with more than one external leg. According to Eq. (2.16) the tadpole renormalization conditionT i = 0 does not modify connected Green's functions in the FJ Tadpole Scheme since ∆v can be freely chosen, in particular, such that the tadpole equations (2.13) are fulfilled. This has interesting consequences for the interpretation of the tadpole counterterms. For example, using that the expression (2.9) is independent of ∆v, we conclude that the one-loop two-point tadpole counterterm derived from Eq. (2.4) obeys independently of the nature of the external particle(s). This can also be seen directly at one-loop order by computing the two-point tadpole contribution t ij which can be derived from Eq. (2.19).
To this end, we assume a typical scalar potential V in the Lagrangian (2.1) without derivative interactions. Expanding t ij to first order in ∆v and using that ∆v acts as a field shift, we obtain t ij = F.T. n ∆v n ∂δ 2 S ∂∆v n δϕ i δϕ j ϕ=0,∆v=0 where we use that Γ is given by the action S at tree-level 3 and F.T. denotes the Fourier transform that translates Green's functions from configuration space to momentum space. Defining the mass squared matrix of the scalar fields the explicit tadpole counterterm to the two-point function reads (2.23) 3 The contribution (2.21) is a higher-order contribution because ∆v is identified with a higher-order correction.

6
For the tadpole counterterm, we find Inserting this into Eq. (2.23), yields (2.26) Therefore, the tadpole counterterms mimic the contribution of tadpole diagrams T i , once T i is identified with −t i . We conclude that the FJ Tadpole Scheme is equivalent to a scheme where tadpoles are not renormalized, which corresponds to setting ∆v to zero in Eq. (2.2) and computing all tadpole diagrams explicitly. The consistent use of the FJ Tadpole Scheme defined in Eqs. (2.2) and (2.5)-(2.7) guarantees the independence of the chosen tadpole renormalization, meaning that the value for any physical quantity is independent of the value of t i and thusT i .
We stress that the shift ∆v i in the vevs is not a parameter of the theory but can be chosen arbitrarily. By solving the tadpole equation (2.13), which can be done order by order in perturbation theory, the shift can be expressed as a function of tadpole counterterms. After spontaneous symmetry breaking, the bare physical parameters like particle masses can be expressed through the theory defining parameters, i.e. the coupling constants in the Higgs potential before spontaneous symmetry breaking. In the FJ Tadpole Scheme, tadpole contributions are never absorbed into the definition of bare physical parameters, which is crucial to assure gauge independence in some renormalization schemes.
Here, we would like to mention a general consequence of BRST invariance [21]: S-matrix elements calculated in terms of the bare theory-defining parameters c j,B of Eq. (2.1) are gauge independent. Thus, renormalization schemes that fix the c j,B in a gauge-independent way lead to a gauge-independent S-matrix. A possible gauge-dependent definition of ∆v does not spoil the gauge independence of the S-matrix, as long as it does not enter the renormalization conditions. However, the latter requirement is violated in popular schemes as detailed below.
In the following, within the FJ Tadpole Scheme we always take advantage of the tadpole renormalization conditionT i = 0, and explicit (counterterm-)tadpoles do not show up. In Section 2.2, we describe the FJ Tadpole Scheme scheme in the SM, and in Section 4.1, it is applied to the 2HDM. For both models, we discuss how the bare physical masses are properly related to the original parameters of the Lagrangian.

The FJ Tadpole Scheme in the SM
In the SM, physical parameters such as the particle masses and the EW couplings are usually renormalized using on-shell and physical renormalization conditions leading to gauge-independent physical observables. 4 Nevertheless, the techniques discussed in the previous section, which result in gauge-independent counterterms to physical parameters in arbitrary renormalization schemes, can already be illustrated in the SM. Following the notation in Ref. [22], the bare Lagrangian for the Higgs field Φ B defined in Eq. (2.1) can be written as (2.27) The Higgs field couples to the gauge bosons through the covariant derivative D µ . The bare Higgs potential is given by with the bare Higgs doublet Φ B defined as . (2.29) We insert the Higgs doublet into the potential and collect the terms linear, V 1 B , and quadratic, V 2 B , in the bare, neutral, scalar Higgs field h B , (2.30) The relation between ∆v and the tadpole counterterm t h is determined according to Eq. (2.5), Using the tree-level condition, ∆v = 0 ⇔ t h = 0, gives the relations between the bare parameters where the last relation defines the bare W-boson mass. The exact form of the shift ∆v in terms of the tadpole counterterm t h can be obtained from Eq. (2.5), which requires the knowledge of the linear term of the Higgs potential (2.30) for ∆v = 0, (2.33) 4 The renormalization of the strong coupling is not influenced by tadpoles at the one-loop level. 8 We can relate ∆v to the tadpole counterterms t h at every order in perturbation theory. For ∆v = ∆v (1) + ∆v (2) + . . . and t (L) h being the L-loop tadpole counterterm corresponding to the SM Higgs boson, we obtain at one-loop order and at two loops. The tadpole counterterm to the two-point function of the SM Higgs boson can be obtained from Eq. (2.6) where λ B , µ 2 B , and v B are replaced according to Eq. (2.32) as before. This yields At one-loop order the bare parameters in Eq. (2.37) can be replaced by renormalized ones. Omitting anything beyond one loop, the two-point tadpole counterterm is given by Using the on-shell condition q 2 = M 2 h,R , where M h,R denotes the renormalized Higgs-boson mass, and the tadpole renormalization conditionT h = 0, the renormalized on-shell two-point function of the Higgs boson readŝ This expression can be used to determine the counterterm of the Higgs-boson mass δM 2 h . We note that at the two-loop order bare parameters need to be expressed in terms of counterterms and renormalized parameters in Eqs. (2.34) and (2.37), omitting any terms beyond two loops. The Higgs-boson self-energy at two loops, focussing on the tadpole dependence, is discussed in App. C.
We note that additional tadpole counterterms are required in the SM for two-and threepoint functions of scalars, vector bosons and fermions. Tadpole counterterms for two-and threepoint functions involving vector bosons originate from the kinetic terms of the SM Higgs sector, while tadpole counterterms to fermion self-energies result from the Yukawa terms of the SM Lagrangian. 9

Gauge independence of physical parameters in the SM
As mentioned at the beginning of Section 2, the physical parameters of the SM are usually renormalized on shell. In this case, gauge dependencies introduced by careless treatments of tadpoles cancel in all renormalized physical quantities. However, when some parameters are renormalized in the MS scheme this is not generally the case. Such problems can originate from gauge-dependent counterterms resulting from a gauge-dependent definition of bare physical parameters. We use the SM to demonstrate potential problems with gauge dependence in renormalization schemes commonly used in the literature. In order to illustrate the effect of different tadpole renormalization schemes on the gauge dependence of the counterterms of physical parameters, we compare the gauge dependence of the Higgs-boson mass counterterm δM 2 h in the scheme described in Ref. [22] and the β h scheme from Ref. [20] to the FJ Tadpole Scheme.
The scheme described in Ref. [22] requires the vev of the bare Higgs-boson field to vanish at one-loop order such that t h is fixed via Eq. (2.13) and thus gauge dependent. At the same time, the bare Higgs-boson mass M 2 h,B is defined as the coefficient of the quadratic term in the Higgs field. As a consequence, no tadpole counterterm t hh appears in the two-point function. However, the so-defined bare Higgs-boson mass, e.g. at one loop depends on the tadpole counterterm t h and thus becomes gauge dependent as well.
The mass counterterm of the Higgs boson, defined as the difference between the bare mass squared M 2 h,B and the renormalized Higgs-boson mass squared M 2 h,R , is determined by requiring that the renormalized self-energy (2.9) vanishes on-shell, i.e. for q 2 = M 2 h,R . Since the renormalized tadpole contributionT (1) h vanishes [see Eq. (2.40)], the Higgs-boson mass counterterm is given by the 1PI contribution The gauge dependence of Σ 1PI hh (M 2 h,R ), which results in a gauge-dependent mass counterterm δM 2 h , can be shown by means of an explicit calculation as in Ref. [23].
In the scheme of Ref. [22] also the bare gauge-boson and fermion masses become gauge dependent, since they are defined using the shifted vev (v B + ∆v). For instance, the bare Wboson mass is given by at one-loop order. The gauge dependence of the Higgs-boson mass counterterm can also be understood from its definition (2.42). As the renormalized mass parameter is identified with the physical mass in the on-shell scheme, which has to be gauge independent, the gauge dependence of a bare parameter must be compensated by the gauge dependence of the counterterm. Using the shorthand notation ∂ ξ for ∂/∂ξ, (2.42) leads to in the R ξ -gauge with gauge parameter ξ. As ∂ ξ M 2 h,R = 0, the gauge dependence of M 2 h,B is directly related to the gauge dependence of δM 2 h . A similar discussion applies to the β h scheme in Ref. [20] which also requires the vev h B to vanish at higher orders and defines the bare masses using the shifted vev, e.g.
at one-loop order. In this scheme the parameter µ B is eliminated from the bare Lagrangian in favour of t h and M 2 h,B , while λ B is expressed in terms of M 2 h,B , M 2 W,B and g B . As a consequence, all tadpoles to 3-point functions are absorbed into the definition of the bare physical parameters. This has the advantage that tadpole counterterms appear exclusively in one-and two-point functions for the Higgs and would-be Goldstone bosons. However, the bare masses become gauge dependent via the dependence on the tadpole t h . The Higgs-boson mass counterterm reads This definition differs from the one resulting from Eq. (2.39) upon imposing the on-shell mass renormalization conditionΣ hh (M 2 h,R ) = 0 by gauge-dependent tadpole terms. In Ref. [20], the β t scheme has been introduced to cure this problem. There, the bare particle masses are defined in terms of the bare vev and are therefore gauge independent. When tadpoles are renormalized requiringT h = 0, the FJ Tadpole Scheme in the SM is equivalent to the β t scheme.
The theory-defining bare parameters of the original Lagrangian, e.g. λ B and µ B are gauge independent by definition. When introducing a new set of bare parameters, these can become gauge dependent if vevs or tadpoles enter their definition. Bare parameters that are defined exclusively by the theory-defining bare parameters remain gauge independent.
In the scheme of Ref. [22] and the β h scheme of Ref. [20] discussed above, all bare particle masses are gauge dependent. However, the gauge dependence cancels in physical quantities since all parameters of the theory are defined by on-shell renormalization conditions. This can be seen as follows: the gauge dependence of the counterterms results from the omission of gauge-dependent tadpole contributions. Since these are momentum independent they cancel in renormalized quantities that are defined by subtracting the same quantity at a fixed point in momentum space. This holds for on-shell schemes or momentum-subtraction schemes but not for MS or MS schemes, where only the divergent parts are subtracted. Since in the scheme of Ref. [22] as well as in the β h scheme of Ref. [20] all renormalization conditions are based on complete subtraction of the relevant vertex functions, the resulting physical quantities and scattering amplitudes are gauge independent. This changes when some parameters are renormalized in the MS scheme, potentially inducing gauge dependence in the S-matrix if applied to gauge-dependent bare parameters.
For later convenience we describe a simple way to construct the different tadpole schemes from the original bare Lagrangian (2.2) or (2.27) augmented by gauge, fermion and Yukawa terms. The starting point is the bare Lagrangian in terms of the theory-defining parameters, i.e. µ 2 B , λ B , g B , . . . for the SM, where the bare scalar fields have been shifted by an independent parameter v B (which is not yet fixed by a minimum condition) and vanishing ∆v. Then, the tadpole renormalization in the different schemes at one-loop order can be introduced by shifting the bare parameters as follows:

Scheme 1
The bare Lagrangian in the scheme of Ref. [22] is obtained upon performing the shifts

Scheme 2
The bare Lagrangian in the β h scheme of Ref. [20] results from the shifts Only after these shifts the vev v B is fixed by minimizing the scalar potential for t h = 0 and thus related to µ 2 B and λ B . We have shown that the FJ Tadpole Scheme is the natural scheme for dealing with the tadpoles, as it prevents that tadpoles are absorbed into the definition of bare parameters. Moreover, the tadpole renormalization conditionT = 0 is very useful since no explicit tadpole loop contributions have to be computed. We stress that the presented tadpole renormalization procedure is general and not restricted to the SM or the 2HDM.

Two-Higgs-doublet model-Lagrangian and fields
In this section, we review the definition of the Lagrangian of the 2HDM. We restrict ourselves to the case of a CP-conserving type-II 2HDM with a softly broken Z 2 symmetry. For a comprehensive introduction to the 2HDM we refer to e.g. Refs. [3,4].

Fields and potential in the symmetric basis
Let Φ i denote the i-th Higgs doublet with i = 1, 2 defined by (3.1) The most general potential of the 2HDM has 3 quadratic and 7 quartic products of Higgs doublets, each coming with a real or complex coupling constant. Requiring CP conservation and Z 2 symmetry (Φ 1 → −Φ 1 ,Φ 2 → Φ 2 ) simplifies the Lagrangian resulting in five real couplings λ 1 . . . λ 5 and two real mass parameters m 2 1 and m 2 2 . Soft breaking of the Z 2 symmetry allows for the third mass parameter m 2 12 . Since the theory is spontaneously broken, we assign two vacuum expectation values v 1 and v 2 which, under the same symmetry restriction, can be chosen to be real. The most general potential is then given by

Fields and potential in the mass eigenbasis
After spontaneous symmetry breaking the eight degrees of freedom in the doublets (3.1) split into three would-be Goldstone bosons G 0 and G ± and five physical Higgs bosons H l , H h , H a , H ± . In order to identify the mass eigenstates, the part of the Lagrangian quadratic in the fields needs to be diagonalized. The mixing angle β is introduced to separate would-be Goldstone bosons from charged and pseudoscalar physical Higgs fields, and the angle α is required to diagonalize the neutral Higgs sector. With the rotation matrices the mass eigenstates of Higgs-and would-be-Goldstone-boson fields are obtained by the following transformations for a suitable choice of α and β. The Higgs sector is coupled to the gauge sector by means of covariant derivatives. Identifying the mass eigenstates of the vector bosons, one obtains the well-known tree-level relations in the 2HDM where g and g denote the weak isospin and hypercharge gauge couplings, and M W and M Z the W-and Z-boson masses, respectively. The mixing angle β is related to the ratio of vevs according to t β ≡ tan β = v 2 /v 1 . The angle α is chosen such that it diagonalizes the symmetric mass-squared matrix defined by where V is the potential in Eq. (3.2). The solution reads [3] sin 2α = 2M 12 13 The parameters of the Higgs potential can then be substituted for physical parameters after SSB and after diagonalizing the neutral Higgs sector. The minimum conditions for the scalar potential, ρ i = 0, read where the vev v has been substituted using Eq. (3.5).

Yukawa Lagrangian for the type-II 2HDM
In the type-II 2HDM, the up-type quarks couple to Φ 2 , while the down-type quarks and leptons couple to Φ 1 . This corresponds to the Higgs sector in the MSSM, but here, it is realized by the discrete whereΦ 2 is the charge-conjugated Higgs doublet of Φ 2 . Neglecting flavour mixing, the coefficients are expressed by the fermion masses m d , m u and m l , and the mixing angle β, Again, the vev v has been substituted using Eq. (3.5).

Physical parameters
In the mass eigenbasis the physical parameters resulting from the Higgs sector are identified with the Higgs-boson masses, M H l (light Higgs boson), M H h (heavy Higgs boson), M Ha (pseudoscalar Higgs boson), M H ± (charged Higgs boson), the two mixing angles α and β, the soft-Z 2 -breaking scale M sb , and the vacuum expectation value v. The mass of the light Higgs boson is commonly identified with the mass of the observed Higgs boson, and v also keeps its SM value being directly related to the W-boson mass M W . This leaves a total of six new parameters compared to the SM. This identification allows to translate the parameters in the symmetric basis to the mass eigenbasis Note that the vevs v 1 and v 2 are no independent physical parameters. In the following, we choose a more natural representation for the angles in view of the alignment limit [3] α, β → c αβ := cos(α − β), t β := tan β, (3.14) which can be achieved by using simple trigonometric identities. 5 We have chosen the sign convention for the angles in such a way that the alignment limit is reached by

Renormalization conditions in the 2HDM
As argued in Section 2, the 2HDM is an example of a theory with new parameters, namely the mixing angles α and β and the soft-Z 2 -breaking scale M sb , whose on-shell renormalization through vertex functions would introduce a process dependence or would be plagued by IR singularities. This has been discussed for the decays of heavy scalar and charged Higgs bosons in Ref. [11] and for the renormalization of the mixing angle β in the context of the MSSM in Ref. [6]. Therefore, an MS renormalization is advantageous. It requires, however, care in the treatment of tadpoles to assure the gauge independence of the bare physical parameters of the theory and thereby of the S-matrix. This is guaranteed by the FJ Tadpole Scheme presented in Section 2. In this section, this scheme is applied to the 2HDM. The corresponding renormalization conditions are listed in Sections 4.3-4.5, including the tadpole counterterms which are essential for the gauge independence of the expressions. The bare parameters split into the finite, renormalized parameters and counterterms, where f stands for any fermion. In the SM, only the Z-boson field Z and the photon field A mix and require the introduction of renormalization matrices. In the 2HDM, we introduce additional renormalization matrices for the mixing of the two neutral scalars H l and H h , the pseudo scalars G 0 and H a , and the charged scalars G ± and H ± . The complete field renormalization is given by The fermion field renormalization is defined as for left-handed (L) and right-handed (R) fermions, where we neglect fermion mixing.

The FJ Tadpole Scheme applied to the 2HDM
The 2HDM as presented in the previous section contains two Higgs doublets with the corresponding two vevs, such that Eq. As in the SM, we use Eq. (2.5) to obtain ∆v 1 and ∆v 2 expressed by the L-loop tadpole counterterms, but instead of calculating the tadpoles in the generic basis, we define the vevs in terms of the tadpole counterterms associated to the physical Higgs fields H l and H h . Of course, the result does not depend on the choice of parametrization. In the 2HDM, the tadpole counterterms are defined as At tree level, the tadpole counterterms t H l and t H h vanish, such that ∆v 1 = ∆v 2 = 0. This provides the conditions (3.8) for the potential minimum at tree level, and the relations between the generic and the physical Higgs basis (3.10) for bare quantities. Thus, the bare parameters in the symmetric basis are expressed by the bare parameters in the physical basis. Linearizing Eq. (4.5) by using the expansion (2.14), we can solve for ∆v 2 . The results for ∆v (1) 1 and ∆v (1) 2 simplify after using the potential minimum conditions at lowest order and the relation between the parameters in the generic and physical basis.
Evaluating the linearized versions of Eq. (4.5) for bare physical parameters, we obtain the one-loop expressions ∆v (1) Just as in the SM, tadpole counterterms arise from all terms in the Lagrangian that depend on the vevs. This results in tadpole counterterms to two-and three-point functions involving scalars and vector bosons as well as to fermionic two-point functions.
In the following, we assume that the tadpole counterterms are fixed according to Eq. (2.13). Explicit results for the tadpoles T H l and T H h in the 2HDM in the R ξ -gauge are listed in App. A.

Renormalized two-point functions
Using Eq. (2.9) and the conditionT i = 0, the renormalized self-energiesΣ(q 2 ) for vector bosons are given bŷ for V = {W, Z, A}, with the 1PI contributions Σ 1PI,T V V , and for the mixing of photons and Z bosons. The scalar sector works similarly, witĥ for the mixing of the scalar fields, where SS = {G 0 H a , G ± H ± , H h H l }. The renormalized scalarvector-boson mixing energy readŝ where V S = {W ± G ∓ , W ± H ∓ , ZG 0 , ZH a }. Defining the helicity projectors the renormalized fermionic self-energies can be decomposed into covariantŝ which are given bŷ (4.14) We omit the renormalization of Faddeev-Popov ghosts which are not needed for the discussion of the processes under consideration (see Section 6) and in general not at one-loop order. As a result of using the FJ Tadpole Scheme, all self-energies involving massive particles receive tadpole contributions. These tadpole contributions assure the gauge independence of the on-shell selfenergies. Results for the tadpole contributions entering the renormalized self-energies in terms of the tadpole counterterms t H l and t H h are provided in the 't Hooft-Feynman gauge in App. B.

Mass and field renormalization conditions
In the complex-mass scheme 6 [24][25][26], the scalar and vector-boson mass and field renormalization constants are derived from the conditionŝ For fermions, the renormalization conditions are given by Inserting the expressions (4.7)-(4.14) into the renormalization conditions (4.15)-(4.17), we obtain the mass and field renormalization constants in terms of the 1PI self-energy and tadpole contributions.

Renormalization of the electroweak coupling
The electromagnetic coupling e as well as the weak coupling g can be related to the fine-structure constant α (not to be confused with the mixing angle of the neutral, scalar Higgs bosons) where we define the weak mixing angle in the on-shell scheme as Renormalizing the electromagnetic coupling in the Thomson limit and using a Ward identity leads to [22,27] (4.20) Since the counterterm δe does not receive any tadpole contributions, the 1PI self-energies can be replaced by the full self-energies in Eq. (4.20).
In the G F scheme, the fine-structure constant is expressed by the Fermi coupling constant G F , using the well-known relation where ∆r contains the EW corrections to muon decay. The correction term ∆r depends on the on-shell photon self-energy Σ 1PI,T AA (0), the on-shell Z-boson self-energy Σ 1PI,T ZZ (M 2 Z ), the Wboson self-energy Σ 1PI,T W W at q 2 = 0 and q 2 = M 2 W , the photon-Z-boson mixing energy Σ 1PI,T AZ (0), and explicit vertex and box contributions to the muon decay [28][29][30]. Since the tadpole terms cancel within ∆r, this quantity takes the same form in terms of the full self-energies or their 1PI parts. Under the assumption that the couplings of the Higgs bosons to electrons and muons are negligible, only the self-energies are modified in the 2HDM, while the vertex and box contributions to the muon decay remain the same as in the SM.
In the G F scheme, the renormalization constant for the electromagnetic coupling reads The angle β is renormalized using MS subtraction for the process H a → τ − τ + , where P.P. denotes the projection onto the pole part including the generic finite parts that are subtracted within the MS scheme, i.e. the terms proportional to with the space-time dimension D. The corresponding counterterm explicitly reads (4.25) Since there are no explicit tadpoles for this vertex, it follows that there are no tadpole counterterms in the FJ Tadpole Scheme. The renormalization condition (4.23) determines δβ MS . It has been remarked before (e.g. Refs. [11,13]) that the relation holds at one-loop order, and we have explicitly verified this in the general R ξ -gauge. This relation has also been used as a renormalization condition [11,13]. It is particularly useful because it is valid in any of the schemes which we consider in Section 5. We verified by explicit calculation in the R ξ -gauge that δβ MS is gauge independent in the FJ Tadpole Scheme. The gauge dependence of δβ MS in Schemes 1 and 2 is discussed in App. D.

Mixing angle α
The angle α is renormalized using MS subtraction in the process H l → τ − τ + , The corresponding counterterm is given by Again, there is no tadpole dependence, and the renormalization condition (4.27) determines δα MS . Similarly to Eq. (4.26) the relation 20 is valid at one-loop order, which we have explicitly verified in the R ξ -gauge. Moreover, we have checked by explicit calculation in the R ξ -gauge that δα MS is gauge independent in the FJ Tadpole Scheme but gauge dependent in Schemes 1 and 2. In addition, we have verified that the renormalized vertex H l τ − τ + is gauge independent in the FJ Tadpole Scheme, while it is gauge dependent in Schemes 1 and 2.

Soft-breaking scale M sb
The parameter M sb is renormalized using the MS subtraction of the process H h → H l H l , The dependence of this vertex on δM sb and the tadpole counterterm where (...) stands for other counterterms.

Discussion of gauge dependence
In this section, we discuss the gauge dependence of S-matrix elements assuming that the renormalization conditions listed in Section 4 are employed. We show that tadpole renormalization schemes that are commonly used in literature in combination with MS renormalization lead to gauge-dependent predictions, while the use of the FJ Tadpole Scheme ensures gauge independence.

Gauge-fixing Lagrangian
To verify gauge independence of S-matrix elements and counterterms of physical parameters in the FJ Tadpole Scheme, we use a general R ξ -gauge. The corresponding gauge-fixing Lagrangian is given by We do not renormalize the gauge-fixing Lagrangian, i.e. we write it directly in terms of renormalized fields, which is sufficient to assure that all S-matrix elements are finite [31,32]. To compensate the unphysical components in L GF , Faddeev-Popov ghosts are introduced as usual.

Characterizing different schemes
In the literature different tadpole renormalization schemes are employed. In order to efficiently generate the tadpole counterterms we follow the recipe presented at the end of Section 2.3 for the SM. We start from the tree-level Lagrangian (2.2) in the symmetric basis in terms of the theory-defining parameters m 2 i,B , i = 1, 2, m 2 12,B , λ 2 j,B , j = 1, . . . , 5, where the fields have been shifted by independent parameters v i,B . Then we perform the shifts of the parameters as defined below. Thereafter, the vevs v i,B are determined at leading order, and the bare physical basis is introduced by using the tree-level relations (3.8)-(3.10), i.e. for t H l = 0 = t H h . Finally, the bare parameters are renormalized according to Eq. (4.1).
The tadpole renormalization in the different schemes can be generated by shifting the corresponding bare parameters as follows:

Scheme 1
A commonly used renormalization scheme for the SM was proposed in Ref. [22]. There, the bare physical masses are defined as the coefficients of the quadratic terms in the fields, and the tadpoles are the coefficients of the terms linear in the fields. Applying this definition to the 2HDM, we can construct the corresponding Lagrangian by a shift in the bare parameters as One can verify that the prescription (5.3) leads to the tadpole equations (4.5) in the 2HDM. Note that in the alignment limit, the SM tadpoles (see App. A in Ref. [22]) are reproduced.

Scheme 2
In the β h scheme of Ref. [20], the mass parameters in the Higgs potential are eliminated in favour of explicit tadpoles, while the quartic Higgs couplings λ i are kept fixed. Thus, no tadpole counterterm contributions appear in the triple and quartic vertices between scalars, but the mass parameters of the Higgs potential and thus the two-point functions are shifted by tadpole counterterms, For explicit computations, Scheme 2 is very simple because tadpole counterterms appear only in two-point functions. This scheme is widely used, e.g. in the 2HDM [8][9][10]13,33] and in the MSSM [5][6][7]. In contrast, in Scheme 1 two-point functions do not receive tadpole counterterms due to the definition of the bare masses in that scheme.

Scheme 3
As described in detail in Section 2.1, in the FJ Tadpole Scheme, the vevs are replaced by (v 1,B + ∆v 1 ) and (v 2,B + ∆v 2 ), which corresponds to the following shift This prescription has to be applied to the full Lagrangian and is not restricted to the Higgs potential.
We stress again, as we have shown in Section 2.2, that the bare parameters of the theory are shifted by (gauge-dependent) tadpole contributions in Schemes 1 and 2, as opposed to the prescription of the FJ Tadpole Scheme (5.5), where only the unphysical vevs receive a shift.

Differences of counterterms in different renormalization schemes
Employing different schemes leads to different expressions for the counterterms. Since we are mainly interested in the changes of amplitudes between different tadpole renormalization schemes, we compare counterterms in the different schemes. We name the schemes as in the previous section, i.e. Scheme 1 for the scheme employed in Ref. [22] and Scheme 2 for the β h scheme of Ref. [20]. The FJ Tadpole Scheme is referred to as Scheme 3. We generically label the difference in the schemes for a counterterm δc i by where the ∆ i describe the difference of Scheme i with respect to the Scheme 3.
In the following, we list the results for the counterterm parameters. Thereby, we make use of results for tadpoles listed in Apps. A and B. As a first result, we note that the counterterms of couplings in the SM are not affected by the choice of the tadpole renormalization scheme, i.e.
However, the masses of all fermions and gauge bosons change equally for i = 1, 2 as which is easily derived because neither in Scheme 1 nor in Scheme 2 there are tadpole contributions to two-point functions of fermions and gauge bosons. Therefore, the difference is the full tadpole dependence of these two-point functions in the FJ Tadpole Scheme obtained from Eq. (5.5). The results for the scalar fields are more complicated but not needed in the following. For Scheme 1, the difference is again given by the full tadpole dependence in the FJ Tadpole Scheme, which can be found in App. A. In the FJ Tadpole Scheme, the mass counterterms are gauge independent by definition, which we have verified in a general R ξ -gauge. Consequently, the mass counterterms in Schemes 1 and 2 are gauge dependent, and their gauge dependence is given by the gauge dependence of the corresponding tadpole counterterms.
Next, we give the results for the new parameters in the 2HDM. Since those parameters are renormalized in the MS scheme, we only need to study the UV-divergent parts of vertex functions and can use the Eqs. (4.26) and (4.29), which hold in any of the presented schemes. For β, we obtain where "MS" denotes the UV-divergent part of the corresponding expression together with the finite terms in the MS-scheme according to (4.24). In the first step, we use Eq. (4.26). The second step can be derived by solving the renormalization conditions for the relevant mixing energies where we omitted any explicit tadpoles because ofT i = 0. Since the self-energy diagrams do not depend on the scheme, the scheme-dependent divergence of the tadpole counterterms has to cancel against the scheme-dependent divergence of the non-diagonal field renormalization constants, and for i = 1, 2 we obtain Therefore, the scheme dependence of δβ MS is given by the one of the tadpole contribution t G 0 Ha of Eq. (5.9). The explicit results for t G 0 Ha in the FJ Tadpole Scheme are listed in App. B, and those for Schemes 1 and 2 are given by 13) and hence While the change in δβ MS at one-loop order is independent of the gauge parameters in the usual R ξ -gauge and in their generalizations to non-linear gauges, we show in App. A that it is nevertheless already gauge dependent at the one-loop level in the 2HDM. We expect that this applies as well to the MSSM, where it is known that δβ MS becomes gauge dependent at two loops [34]. For the difference in the counterterms to the mixing angle α, we obtain and Here we used Eq. (4.29) and the antisymmetry of ∆ 1 δZ MS H l H h , which can be derived similarly as the one of ∆ 1,2 δZ MS HaG 0 above. The result for ∆ 1 δα MS can be expressed by the tadpole counterterm in the FJ Tadpole Scheme only, because Scheme 1 [see Eq. (5.3)] does not induce tadpole counterterms for two-point functions that do not involve external would-be Goldstone bosons.
The differences ∆ 1 δα MS and ∆ 2 δα MS are both gauge dependent at one-loop order, which can be seen by inserting the explicit expressions for the tadpoles from App. A in the R ξ -gauge. This result is used in the next section to demonstrate the gauge dependence of S-matrix elements in Schemes 1 and 2.

The H l τ + τ − vertex
We study the renormalized H l τ + τ − vertex for the different tadpole renormalization schemes defined above. In this section, we assume that the renormalized tadpole terms vanish,T (1) H l = 0 andT (1) H h = 0. Then, the shifts in the bare parameters that originate from tadpole counterterms t As the tadpole renormalization schemes do not modify the bare Lagrangian, the bare loop amplitudes are not altered. However, the finite parts of the counterterms are affected, and thus receive gauge dependencies, as we demonstrate in the following. In particular, we show that the change in the renormalized vertex function is gauge dependent in the R ξ -gauge, i.e.
which has been verified by direct computation.

25
The relevant Feynman rules read Computing the difference of the renormalized vertex function in different tadpole schemes yields The terms can be split into two parts which are separately UV finite, thus allowing for a simple interpretation where we used Eqs.  The FJ Tadpole Scheme yields gauge-independent predictions for the decay rate H l → τ + τ − , whereas in Schemes 1 and 2 the prediction is gauge dependent. This has been confirmed via explicit calculation of the S-matrix element in the R ξ -gauge.

26
At one-loop order, the results for the FJ Tadpole Scheme can be obtained from Schemes 1 and 2 via the mapping It is interesting to mention that the "Tadpole scheme" introduced in Ref. [6] for the renormalization of t β in the MSSM is equivalent to the FJ Tadpole Scheme applied to the MSSM combined with MS subtraction for t β . Indeed for the MSSM the finite shift δt fin β defined in Eq. (43) of Ref. [6] corresponds exactly to the shift of δβ MS 2 in Eq. (5.24), which translates the popular Scheme 2 to the FJ Tadpole Scheme. While in the FJ Tadpole Scheme the MS subtracted t β is directly gauge independent, an additional finite renormalization is required in Scheme 2 to restore the gauge independence after MS subtraction.

The ZZH h vertex
In this section, we present the finite correction of the ZZH h vertex due to the tadpole scheme. We obtain formally analogous results as in the previous section. The following Feynman rules were used The calculation proceeds as in Section 5.4 except that one has to take into account a tadpole contribution which is given by With the same line of arguments as in the previous section we obtain the difference of the renormalized vertex in different tadpole schemes as The gauge independence of the renormalized ZZH h vertex has been verified in the FJ Tadpole Scheme by explicit computation in the R ξ -gauge. In this way it has also been confirmed that this vertex is gauge dependent in Scheme 1. Schemes 1 and 2 can be mapped to the FJ Tadpole Scheme by a redefinition of α and β via Eq. (5.24). It is expected that the same is true for other vertices which are sensitive to the renormalization of α, β, and SM parameters, but not to M sb . Since the mapping (5.24) is gauge dependent, the renormalized ZZH h vertex becomes gauge dependent in Schemes 1 and 2.  Table 1: 2HDM benchmark points in the alignment limit, i.e. s αβ → −1, c αβ → 0, taken from Ref. [35]. The parameter M sb depends on the other parameters and is given for convenience.

Electroweak NLO corrections to Higgs-boson production processes in the 2HDM
In this section, we analyze the EW NLO corrections for two Higgs-boson production channels in the 2HDM. First, in Section 6.1, we discuss results for the production of a light SM-like Higgs boson produced through gluon fusion for scenarios in the alignment limit. A more detailed description of the implementation of this process and results for the production of a light or a heavy neutral Higgs boson for the case of non-alignment will be presented elsewhere. In Section 6.2, we provide results for the production of a light SM-like Higgs boson in vector-boson fusion at NLO. Also here, a more detailed study including the description of the implementation of the process will be published separately.
In both processes all external particles are SM particles such that the new Higgs bosons only appear as virtual particles in the loops. In both cases, we apply the renormalization scheme defined in Section 4 7 and discuss the size of the EW corrections. We study the dependence on the renormalization scale that appears owing to the MS renormalization of the mixing angles α and β and analyze the decoupling of the new (heavy) Higgs particles. Besides investigating scenarios close to the decoupling limit, we provide results for selected benchmark points from Refs. [35,36]. The benchmark points BP21A-D, BP22A, and BP43-5 were originally designed for the study of exotic Higgs-boson decays, the points BP3A1 and BP3B1-2 for a successful EW baryogenesis and the points a-1 and b-1 for Higgs-boson pair production. All benchmark points fulfil theoretical constraints from vacuum stability and perturbativity as well as experimental constraints in flavour physics, EW precision measurements, and direct searches. In Table 1 we list the benchmark points in the alignment limit, which we study in gluon fusion and Higgs strahlung. In Table 2 we provide benchmark scenarios that are not in the alignment limit and which we study in Higgs strahlung only.
For the numerical evaluation of the two Higgs-boson production processes we use the following values for the SM input parameters [37]:  Table 2: 2HDM benchmark points outside the alignment limit taken from Ref. [36] (a-1, b-1) and Ref. [35]. The parameter M sb depends on the other parameters and is given for convenience.

Higgs-boson production in gluon fusion
Higgs-boson production through gluon fusion is a loop-induced process, i.e. its LO contribution appears at the one-loop level. Despite its loop suppression, it is the dominant Higgs-boson production mechanism in the SM at the LHC. Since the Yukawa couplings of the Higgs boson to fermions are proportional to the fermion mass, the dominant contribution arises from topquark loops. Treating all other fermions as massless, the LO partonic cross sectionσ for SM Higgs-boson production is generated only via a top-quark loop.
In the SM, the QCD corrections to Higgs-boson production in gluon fusion are known up to N 3 LO and are large [38][39][40][41][42]. The complete NLO EW corrections have been calculated in Refs. [43,44] and are also sizable. EW radiative corrections may significantly change a process, if BSM particles propagate in the loop. In Refs. [45,46], for example, the influence of a fourth generation of heavy fermions on the EW corrections to Higgs-boson production in gluon fusion has been discussed, and the EW corrections turned out to be large. In the following, we present the behaviour of the NLO EW corrections to this Higgs-boson production channel in the alignment limit of the 2HDM of type II, where the light, neutral Higgs boson H l becomes SM-like. All results are calculated in the FJ Tadpole Scheme with the renormalization conditions given in Section 4 with the exception of the top-quark mass which has been renormalized in the on-shell scheme for gluon fusion.
The coupling of the light neutral Higgs boson to top quarks in the 2HDM of type II is given by the H l tt vertex: In the alignment limit (c αβ = 0, s αβ = −1) the coupling of the light neutral Higgs boson to uptype fermions equals the one in the SM. Therefore, the LO production cross section of the light neutral Higgs boson through gluon fusion in the 2HDM is the same as in the SM, and the QCD corrections do not change. For small t β the alignment limit is reached slower and one speaks of a delayed decoupling [3]. Without alignment, the LO cross section changes only by the factor (c αβ /t β − s αβ ) 2 , such that the relative QCD corrections stay the same. In the alignment limit, the t β dependence of the process disappears at LO, but survives in the NLO EW corrections. The derivation of the counterterms for the NLO calculation requires special care. The fact that the alignment limit implies c αβ = 0 (and consequently s αβ = −1), but does not affect t β , leads to an explicit dependence of the H l tt counterterm on t β , δα, and δβ: As a result, the NLO corrections to this process are still scale dependent despite of the alignment limit. The scale dependence originates from the renormalization of α and β in the MS scheme. The calculation of the NLO EW corrections proceeds in several steps. First, the 2HDM Feynman rules are derived using FeynRules [47]. These are used in the code QGS in order to construct the amplitudes based on Feynman diagrams generated with QGRAF [48]. The program QGS is an extension of GraphShot [49], which has been used to accomplish the corresponding SM calculation and performs the algebraic manipulations of the amplitudes with Form [50]. The reducible scalar products are removed, the symmetries are taken into account in order to reduce the number of loop integrals, the UV renormalization as well as the cancellation of collinear logarithms is performed analytically, and finally the remaining finite integrals are mapped onto form factors. The latter are evaluated numerically with Fortran routines. In the alignment limit of the 2HDM the same types of Feynman integrals arise as in the SM calculation of Refs. [43,44] such that we can employ the same Fortran library for their numerical evaluation. For the numerical evaluation of the two-loop massive diagrams the library uses the methods of Refs. [51,52] for self-energies and of Refs. [44,[53][54][55] for vertex functions.
The NLO EW corrections to the partonic cross section are expressed as percentage correction δ NLO EW relative to the LO result,σ =σ LO +σ NLO =σ LO (1 + δ NLO EW ). (6.4) In order to study the scale dependence we consider the following scenario: We choose tan β = 2 and M * = 700 GeV as a typical mass scale for the new degrees of freedom. The soft-breaking scale M sb and the masses of all heavy Higgs bosons, except for the heavy, neutral one, are set equal to M * . By allowing for a different mass value for the heavy, neutral Higgs boson, we find enhanced scale-dependent logarithms in the alignment limit. This can be seen from the analytical expression for the scale-dependent part of the relative correction to the LO matrixelement squared In Fig. 1 we show the percentage correction δ NLO EW as a function of the heavy, neutral Higgsboson mass M H h for the three different values of the renormalization scale µ and compare the EW corrections of the 2HDM to those in the SM. For a small mass splitting, i.e. for a heavy, neutral Higgs-boson mass M H h in the vicinity of M * = 700 GeV, the size of the corrections is comparable to the one in the SM, and the scale dependence is small, i.e. perturbation theory is well-behaved and higher-order EW corrections can be expected to be small. For large mass splittings, e.g. for a heavy, neutral Higgs-boson mass M H h that deviates significantly from the mass values of the other heavy Higgs bosons and thus from M * = 700 GeV, the scale dependence becomes large. This indicates large uncertainties owing to unknown higher-order corrections, which signals the breakdown of the perturbative expansion and the onset of a non-perturbative regime. This behaviour is expected, since the mass splitting of the heavy Higgs bosons is restricted by perturbativity. The parameters λ i in Eq. (3.2) have to fulfil the condition |λ i | O (1) in order to ensure that the Higgs-boson sector does not become strongly coupled [3]. This is important to maintain tree-level unitarity [56]. The requirement |λ i | O (1) leads to a bound on the mass splitting [3] of where v ≈ 246 GeV is the SM vev. Using this information as well as the knowledge of the NLO EW corrections, we study the region of M H h in Fig. 1 Perturbativity should be realized if the parameter f is sufficiently smaller than one. In Fig. 1 the M H h region corresponding to f > 0.4 is marked in light grey, and the region f > 0.5 in dark grey. The scale dependence can be used to estimate uncertainties from unknown higher-order corrections and provides useful means to determine the onset of the non-perturbative regime. This is supported by Fig. 1, where the scale dependence becomes stronger when f becomes large. We conclude that the scale dependence introduced by the MS renormalization of the mixing angles α and β is less than a percent, as long as perturbativity is not violated, i.e. f < 0.5. Thus, the MS renormalization of α and β provides stable results for sound scenarios in the perturbative regime.
In Fig. 2, we analyze the decoupling of the heavy Higgs-boson sector for different values of f . We vary the mass scale M * and choose the masses of the heavy Higgs bosons as  Table 3 we present the relative NLO corrections for the benchmark points of Table 1 which fulfil perturbativity in the sense that |λ i | < 4π [35]. For all scenarios, we observe  large NLO EW corrections on top of the SM value of 5.1%. Owing to the large corrections, it should be possible to exclude these scenarios at the LHC, as soon as computations for the relevant decay channels like H l → γγ are available at the same order in the weak coupling.
The scale uncertainty turns out to be at the level of ±10% for the benchmark points BP21A, BP21B and BP3A1, but small for BP21C and BP21D. The large scale dependencies are due to rather large values of the λ i in these benchmark scenarios, the largest values for |λ i | ranging between 3.7 and 7.7. The results from Table 3 can be understood from the analytic expression for the scale dependence in Eq. (6.5). For BP21A and BP21B, M H h ≈ M sb and, thus, basically only the m t -dependent terms in the last line of Eq. (6.5) contribute. The scale dependence is enhanced by the factor (M 2

Higgs production in association with a weak boson
Besides the gluon-fusion channel and the vector-boson fusion channel, the associated Higgs production with a vector boson, also called Higgs strahlung, is used to study the properties of the Higgs boson. In this section, we focus on this process which allows in particular to measure the decay mode H → bb and to study BSM physics in the V V H vertex. There has been enormous progress in higher-order calculations to Higgs strahlung in the SM. The QCD corrections are known up to NNLO for the inclusive cross section [57][58][59] as well as for differential cross sections [60,61]. The NLO EW corrections were first computed in Ref. [62] for stable vector bosons. Meanwhile public codes are available including the vector-boson decays, e.g. V2HV [63], MCFM [64], HAWK2.0 [65] and vh@nnlo [66], allowing to study any final state in this process class at NLO QCD and EW and also partially at NNLO QCD. Higgs strahlung has also been investigated in the 2HDM [67], where the ratio of inclusive WH and ZH production for light and heavy Higgs bosons has been studied and the impact of type-I and type-II Yukawa couplings to the SM Higgs production has been analyzed including all available and numerically relevant contributions.
The following analysis is restricted to the case of two charged leptons in the final state, pp → Hl + l − + X. For massless leptons one has to be careful with final-state collinear radiation, which requires special treatment (see e.g. Ref. [68]). We do not recombine collinear photons and leptons and assume that the leptons can be perfectly isolated, which is justified for a pair of muons in the final state. We employ the cuts used in the analysis of Ref. [69], i.e. we require the muons to • have transverse momentum p l T > 20 GeV for l = µ + , µ − , • be central with rapidity |η l | < 2.4 for l = µ + , µ − , • have a pair invariant mass m ll of 75 GeV < m ll < 105 GeV.
In addition, we demand a boosted Z boson with • transverse momentum p Z T > 160 GeV.
All predictions are for the hadronic cross section at the center-of-mass energy of 13 TeV using the NLO PDF set NNPDF2.3 with QED corrections [70]. The numerical results were produced using an extended version of RECOLA [71] and HAWK 2.0 [65]. RECOLA has been used to calculate all needed one-loop S-matrix elements in the 2HDM, and HAWK 2.0 served as integrator for Higgs strahlung.
As in the case of gluon fusion, we discuss the scale dependence in the decoupling limit. In the alignment limit the 2HDM leaves its marks in Higgs strahlung only at NLO, but, in contrast to gluon fusion, Higgs strahlung is scale independent in the alignment limit because the treelevel vertices V V H do not depend on t β but only on c αβ and s αβ (see discussion in Section 6.1). For this reason the analysis has been extended to the decoupling limit with only approximate alignment compatible with perturbative unitarity, i.e. the requirement |λ i | O (1) is extended [3] by We perform an analysis for Higgs strahlung similar to the one for gluon fusion in Fig. 1. In Fig. 3 we present the percentage EW correction δ NLO EW as a function of the heavy, neutral Higgs-boson mass M H h for three different scales µ centered around M * /2. All other parameters are kept fixed, the masses are set to the decoupling scale M * = 700 GeV, and we choose t β = 2. The results are presented for three different scenarios where we investigate the decoupling in terms of c αβ , parametrizing with k = 0.1, 0.25, 0.4. The expected non-perturbative region is shown in dark grey for f > 0.5 and the transition region in bright grey defined by 0.4 ≤ f ≤ 0.5. The first plot in Fig. 3 shows the case k = 0.4, which is at the border of perturbative unitarity independently of f because c αβ is close to its upper limit [see Eq. (6.9)]. This scenario exhibits moderate scale uncertainties of the order of one percent in the perturbative regime. Decreasing k to 0.25 reduces the scale dependence to below one percent in the perturbative region f ≤ 0.4, and for k = 0.1 almost no scale dependence is left. While the considered scenario is not in the decoupling limit, the resulting corrections are nevertheless comparable to those in the SM. The decoupling limit is reached by setting c αβ = 0, where the corrections coincide with those in the SM.
In Tables 4 and 5 we present the results for the benchmark points of Tables 1 and 2. For scenarios in the alignment limit compiled in Table 4 there is no scale dependence, and the differences between EW corrections in the 2HDM and in the SM, where they amount to −12.4%, are typically at the level of one percent.
The scenarios outside the alignment limit shown in Table 5 are more interesting. In the scenarios a-1, b-1, BP43, and BP44, which are close to the alignment limit, the scale variations are small of the order of 0.5%. The corrections are comparable to those in the SM, differing typically at the level of one percent. The scenarios BP3B1, BP3B2, BP45 significantly deviate from the    Table 5: Relative NLO correction δ NLO EW to the integrated cross section of pp → Hl + l − + X for the benchmark points outside the alignment limit of Table 2. The SM correction is −12.4% alignment limit with a mass splitting of more than 200 GeV and exhibit scale uncertainties up to 35%. For all these scenarios we find absolute values of λ of the order of |λ i | /(4π) ≈ 0.3. The scenario BP22A is in the decoupling limit, but does not fulfil condition (6.9). We observe large scale uncertainties of the order of 180% which raises the question of perturbativity of this scenario. In fact, we find λ 2 /(4π) ≈ 1.1 and λ 3 /(4π) ≈ 0.7 for BP22A. Thus, large scale uncertainties signal a breakdown of the perturbative expansion.
In conclusion, as in gluon fusion, violation of perturbative unitarity and large scale dependence are connected. We observe small scale uncertainties of the EW corrections in the 2HDM both in the decoupling limit and for benchmark points that are close to the alignment limit or involve small mass splittings, while respecting perturbative unitarity.

Conclusion
The precise study of theories with extended Higgs sectors is of utmost importance for the investigation of the Higgs sector at the LHC. To this end, NLO corrections of QCD and electroweak origin have to be calculated.
We have proposed a consistent gauge-independent renormalization scheme for the CPconserving 2HDM of type II. While masses are renormalized in the on-shell scheme, the mixing angles of the Higgs sector and the soft-Z 2 -symmetry-breaking scale are renormalized in the MS scheme. To render this approach gauge independent, a consistent treatment of tadpoles is crucial. This is provided by the method proposed by Fleischer and Jegerlehner many years ago for the Standard Model.
We have generalized this method specifically to the 2-Higgs-Doublet Model of type II. We have investigated the difference to popular renormalization schemes used in the literature and clarified their range of applicability. We showed in particular that an MS renormalization of the mixing angles in the extended Higgs sector within popular schemes leads to gauge-dependent predictions in the 2-Higgs-Doublet Model of type II. We expect that this is also the case in the Minimal Supersymmetric Standard Model. The proposed extension of the Fleischer-Jegerlehner tadpole scheme can be straightforwardly applied to more general theories. This opens the way for consistent renormalization prescriptions of theories with more complicated extended Higgs sectors.
We have applied the renormalization scheme to the calculation of NLO EW corrections for Higgs production in gluon fusion and Higgs strahlung and have, in particular, investigated the scale dependence of the corrections and the decoupling of the heavy Higgs bosons within this scheme.
with q µ being the incoming momentum of the corresponding vector boson. The Feynman rules for the tadpole counterterms are obtained by multiplying the tadpole expression with the imaginary unit i.
C Tadpoles in the two-loop Higgs-boson self-energy in the FJ Tadpole Scheme In this appendix, we relate the renormalized two-loop self-energy of the Higgs boson in the SM in the two schemes based onT h = 0 and ∆v = 0. Analogously to Eq. (2.26) at one loop, we show that the two-loop tadpole contributions to the self-energy, which are generated by ∆v in the scheme withT h = 0, reproduce the self-energy in the scheme with ∆v = 0, where unrenormalized tadpoles are explicitly taken into account. The latter situation is given in Eq. (2.11) if the renormalized tadpoles are replaced by unrenormalized ones.
To start with, we note that the 1PI two-loop self-energy and tadpole contributions depend on the tadpole renormalization scheme, more precisely, they differ by ∆v-dependent terms. Having performed the renormalization at one-loop, the 1PI two-loop self-energy diagrams in both schemes are related via where the second diagram on the right-hand side schematically denotes all one-loop self-energy diagrams with an additional insertion of the one-loop two-point tadpole counterterm t (1) hh . Using the one-loop result (2.26) which relates t (1) hh with the bare one-loop tadpole t (1) h , this can be written as Next, we consider the two-loop 1PI tadpole which fixes t (2) h in the scheme whereT h = 0, −t The first equality is the renormalization condition. In the second equality we separate the ∆v-dependent terms, where the second diagram schematically represents all tadpole one-loop diagrams with an additional insertion of t (1) hh . In the third equality we use again the one-loop result (2.26). In the scheme where the tadpoles are renormalized according toT h = 0 the renormalized two-loop self-energy can be expressed exclusively by 1PI contributions and is given byΣ (2) hh q 2 The FJ Tadpole Scheme forT h = 0 includes tadpoles via the ∆v-dependent counterterms. In addition to the ∆v (1) -dependent one-loop counterterms appearing in Eqs. (C.1) and (C.3), the two-loop counterterm induces a further dependence on ∆v (1) and ∆v (2) . In the SM the additional two-loop tadpole counterterms are derived from Eq. (2.37), which can be written as t hh h 2 B = λ hhh,B ∆v + λ hhhh,B 2 (∆v) 2 h 2 B (C.5) upon identifying λ hhh,B and λ hhhh,B as the bare triple and quartic Higgs-boson couplings. The dependence on the two-loop tadpole counterterm t (2) h originates from the term proportional to λ hhh,B ∆v (2) . Using Eqs. (2.35) and (C.3) the contribution of t (2) h can be written as Next, we consider the quadratic 1PI one-loop tadpole contributions which are included in ∆v (2) and ∆v (1) 2 being proportional to λ hhh and λ hhhh , respectively. We identify the two contributions with The last two-loop tadpole counterterms result from products of the one-loop tadpole t (1) h with the counterterms to λ hhh , the Higgs-boson mass [entering via Eq. (2.34)], and the Higgs-boson field-renormalization constant and can be represented as follows: hh q 2 hh q 2 ∆v=0 . (C.9) In the second equation we combine counterterms, evaluated for ∆v = 0, and bare-loop topologies to renormalized objects. The result matches the renormalized two-loop self-energy in Eq. (2.11) when tadpoles are not renormalized, i.e. for ∆v = 0.

D Gauge dependence of β in popular tadpole schemes
In the MSSM, δβ MS as obtained in Scheme 2 is gauge dependent at two loops [34], while it does not dependent on the gauge parameters in the R ξ -gauge at one-loop order. The latter result translates to the 2HDM of type II.
In the 2HDM the apparent gauge independence of δβ MS in the R ξ -gauge at the one-loop level can be understood as follows: Consider the linear combination of Φ 1 and Φ 2 that does not have a vev. Using the explicit rotations in Eq. which enters the shift of ∆ i β MS between the FJ Tadpole Scheme and the popular Schemes 1 and 2 found in Eqs. (5.9) and (5.14). The Higgs field H does neither couple to two gauge bosons nor to two would-be Goldstone bosons, and, moreover, it does not enter the gauge fixing in the R ξ -gauge and thus does not couple to Faddeev-Popov ghost fields. Consequently, there are no gauge-dependent Feynman diagrams for the H tadpole at one-loop order and thus ∆ i δβ MS does not depend on the gauge-parameter in the R ξ -gauge. Since δβ MS is gauge independent in the FJ Tadpole Scheme, this translates to Schemes 1 and 2. This argument can be generalized to non-linear R ξ -gauges at one-loop order. 8 Nevertheless, it is possible to demonstrate the gauge dependence of δβ MS in Schemes 1 and 2 at one-loop order in a suitably chosen gauge. Since H couples to one gauge boson or wouldbe Goldstone boson and H a , we can generate a gauge-dependent contribution to its tadpole by allowing for mixing propagators induced by the gauge fixing. Here, we provide an appropriate gauge-fixing function and prove the gauge dependence of δβ MS via two different approaches. In addition, we show that also in this class of gauges the gauge independence of δβ MS is preserved in the FJ Tadpole Scheme.
The appropriate choice of the gauge-fixing function can be motivated as follows. From the point of view of the FJ Tadpole Scheme the gauge dependence appears in the Schemes 1 [Eq.